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Properties

Label 2-633-633.107-c0-0-2
Degree $2$
Conductor $633$
Sign $0.446 + 0.895i$
Analytic cond. $0.315908$
Root an. cond. $0.562057$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.951 − 1.30i)2-s + (−0.309 + 0.951i)3-s + (−0.500 − 1.53i)4-s + (0.951 − 0.309i)5-s + (0.951 + 1.30i)6-s + (−0.809 + 0.587i)7-s + (−0.951 − 0.309i)8-s + (−0.809 − 0.587i)9-s + (0.499 − 1.53i)10-s + (0.587 − 0.190i)11-s + 1.61·12-s + (0.809 − 0.587i)13-s + 1.61i·14-s + 0.999i·15-s + (−0.363 − 0.5i)17-s + (−1.53 + 0.499i)18-s + ⋯
L(s)  = 1  + (0.951 − 1.30i)2-s + (−0.309 + 0.951i)3-s + (−0.500 − 1.53i)4-s + (0.951 − 0.309i)5-s + (0.951 + 1.30i)6-s + (−0.809 + 0.587i)7-s + (−0.951 − 0.309i)8-s + (−0.809 − 0.587i)9-s + (0.499 − 1.53i)10-s + (0.587 − 0.190i)11-s + 1.61·12-s + (0.809 − 0.587i)13-s + 1.61i·14-s + 0.999i·15-s + (−0.363 − 0.5i)17-s + (−1.53 + 0.499i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.446 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.446 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(633\)    =    \(3 \cdot 211\)
Sign: $0.446 + 0.895i$
Analytic conductor: \(0.315908\)
Root analytic conductor: \(0.562057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{633} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 633,\ (\ :0),\ 0.446 + 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.421956617\)
\(L(\frac12)\) \(\approx\) \(1.421956617\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 - 0.951i)T \)
211 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
5 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
11 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + 1.61T + T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.68639964805696441621612535755, −10.01154221416378726854235507356, −9.358753712442343544619079971037, −8.590991545450273160908410563832, −6.45342988845090995133702912632, −5.68575131936137003730358563787, −5.07175107855881914200910570760, −3.78506068125476433788374115224, −3.17123121487354813499534491146, −1.77678632430681360936367034764, 2.00782974788260612474060929786, 3.64388995303213578945049454162, 4.73132513355169434220410220191, 6.11236780284187645335449138947, 6.35959167860760197244198906526, 6.85357702034016056863822330750, 7.972431015819357613275415493929, 8.890836905227442845437328830976, 10.10256627431058530383188442451, 11.04920192789409707341862641791

Graph of the $Z$-function along the critical line